Assuming that every set is constructible, we find a${\text{\Pi }}_1^1 $maximal cofinitary group of permutations of$\mathbb{N}$which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a${\text{\Pi }}_1^1 $maximal cofinitary group inL.

We study the notion of ????-MAD families where ???? is a Borel ideal on ω. We show that if ???? is any finite or countably iterated Fubini product of the ideal of finite sets Fin, then there are no analytic...